2025/12/16 by N. Richardson, Nicholas J. E. Richardson, Richardson, Nicholas J. E. +4
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Reservoir Engineering and Simulation Methods #Stochastic Gradient Optimization Techniques #cs.NA #math.NA #math.OC
paper · pdf · doi:10.48550/arxiv.2512.13993
openalex publication_date 2025/12/16 · openalex created_date 2025/12/18 · openalex updated_date 2026/07/28
Discretized versions of optimization problems over continuous arguments are routinely solved at a single fine resolution, incurring a per-iteration cost that grows, often superlinearly, with the number of grid points. This paper analyzes a multiscale method that instead solves a hierarchy of increasingly fine dyadic discretizations. Linear interpolation of each coarse solution warm starts the next finer scale using any q-linearly convergent update rule as the inner solver. Each coarse problem is a consistent discretization of the continuous problem. Structural properties such as convexity and smoothness are preserved. For problems with Lipschitz-continuous solutions, two variants of the method converge to the fine-scale solution with explicit error bounds. The fine-scale solution in turn approximates the continuous solution once the grid is sufficiently fine, with quantified constants. The total cost to reach a fixed accuracy is provably lower than that of single-scale optimization whenever the cost of one update grows at least linearly in the problem size. Numerical experiments on probability density demixing problems, including geological survey data, show four- to sevenfold speedups while using a fraction of the memory.